Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix Template:Math is commonly denoted Template:Math, Template:Math, or Template:Math. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.

The determinant of a Template:Math matrix is

<math>\begin{vmatrix} a & b\\c & d \end{vmatrix}=ad-bc,</math>

and the determinant of a Template:Math matrix is

<math> \begin{vmatrix}

a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh.</math>

The determinant of an Template:Math matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of <math>n!</math> (the factorial of Template:Mvar) signed products of matrix entries. It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form with the same determinant, equal to the product of the diagonal entries of the row echelon form.

Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the Template:Math matrices that has the four following properties:

1. The determinant of the identity matrix is Template:Math.
2. The exchange of two rows multiplies the determinant by Template:Math.
3. Multiplying a row by a number multiplies the determinant by this number.
4. Adding a multiple of one row to another row does not change the determinant.

The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.

The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism of a finite-dimensional vector space, the determinant of the matrix that represents it on a basis does not depend on the chosen basis. This allows defining the determinant of a linear endomorphism, which does not depend on the choice of a coordinate system.

Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed Template:Mvar-dimensional volume of a Template:Mvar-dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the Template:Mvar-dimensional volume are transformed under the endomorphism. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.

Two by two matrices

The determinant of a Template:Math matrix <math>\begin{pmatrix} a & b \\c & d \end{pmatrix}</math> is denoted either by "Template:Math" or by vertical bars around the matrix, and is defined as

<math>\det \begin{pmatrix} a & b \\c & d \end{pmatrix} = \begin{vmatrix} a & b \\c & d \end{vmatrix} = ad - bc.</math>

For example,

<math>\det \begin{pmatrix} 3 & 7 \\1 & -4 \end{pmatrix} = \begin{vmatrix} 3 & 7 \\ 1 & {-4} \end{vmatrix} = (3 \cdot (-4)) - (7 \cdot 1) = -19.</math>

First properties

The determinant has several key properties that can be proved by direct evaluation of the definition for <math>2 \times 2</math>-matrices, and that continue to hold for determinants of larger matrices. They are as follows:<ref>Template:Harvnb</ref> first, the determinant of the identity matrix <math>\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}</math> is 1. Second, the determinant is zero if two rows are the same:

<math>\begin{vmatrix} a & b \\ a & b \end{vmatrix} = ab - ba = 0.</math>

This holds similarly if the two columns are the same. Moreover,

<math>\begin{vmatrix}a & b + b' \\ c & d + d' \end{vmatrix} = a(d+d')-(b+b')c = \begin{vmatrix}a & b\\ c & d \end{vmatrix} + \begin{vmatrix}a & b' \\ c & d' \end{vmatrix}.</math>

Finally, if any column is multiplied by some number <math>r</math> (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:

<math>\begin{vmatrix} r \cdot a & b \\ r \cdot c & d \end{vmatrix} = rad - brc = r(ad-bc) = r \cdot \begin{vmatrix} a & b \\c & d \end{vmatrix}.</math>

Geometric meaning

File:Area parallelogram as determinant modified.svg

The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.

If the matrix entries are real numbers, the matrix Template:Math represents the linear map that maps the basis vectors to the columns of Template:Math. The images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the columns of the above matrix is the one with vertices at Template:Math, Template:Math, Template:Math, and Template:Math, as shown in the accompanying diagram.

The absolute value of Template:Math is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by Template:Math.

The absolute value of the determinant together with the sign becomes the signed area of the parallelogram. The signed area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix).

To show that Template:Math is the signed area, one may consider a matrix containing two vectors Template:Math and Template:Math representing the parallelogram's sides. The signed area can be expressed as Template:Math for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g. Template:Math, so that Template:Math becomes the signed area in question, which can be determined by the pattern of the scalar product to be equal to Template:Math according to the following equations:

<math>\text{Signed area} =

 |\boldsymbol{u}|\,|\boldsymbol{v}|\,\sin\,\theta = \left|\boldsymbol{u}^\perp\right|\,\left|\boldsymbol{v}\right|\,\cos\,\theta' =
 \begin{pmatrix} -c \\ a \end{pmatrix} \cdot \begin{pmatrix} b \\ d \end{pmatrix} = ad - bc.

</math>

Thus the determinant gives the area scale factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix preserves area and orientation.

File:Determinant parallelepiped.svg

The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.

If an Template:Math real matrix A is written in terms of its column vectors <math>A = \left[\begin{array}{c|c|c|c} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{array}\right]</math>, then

<math>

 A\begin{pmatrix}1 \\ 0\\ \vdots \\0\end{pmatrix} = \mathbf{a}_1, \quad
 A\begin{pmatrix}0 \\ 1\\ \vdots \\0\end{pmatrix} = \mathbf{a}_2, \quad
                                                          \ldots, \quad
 A\begin{pmatrix}0 \\0 \\ \vdots \\1\end{pmatrix} = \mathbf{a}_n.

</math>

This means that <math>A</math> maps the unit n-cube to the n-dimensional parallelotope defined by the vectors <math>\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n,</math> the region <math>P = \left\{c_1 \mathbf{a}_1 + \cdots + c_n\mathbf{a}_n \mid 0 \leq c_i\leq 1 \ \forall i\right\}</math> (<math display="inline">\forall</math> stands for "for all" as a logical symbol.)

The determinant gives the signed n-dimensional volume of this parallelotope, <math>\det(A) = \pm \text{vol}(P),</math> and hence describes more generally the n-dimensional volume scale factor of the linear transformation produced by A.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This means that A produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.

Definition

Let A be a square matrix with n rows and n columns, so that it can be written as

<math>A = \begin{bmatrix}

 a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
 a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
  \vdots &  \vdots & \ddots &  \vdots \\
 a_{n,1} & a_{n,2} & \cdots & a_{n,n}

\end{bmatrix}.</math>

The entries <math>a_{1,1}</math> etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a commutative ring.

The determinant of A is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:

<math>\begin{vmatrix}

 a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
 a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
  \vdots &  \vdots & \ddots &  \vdots \\
 a_{n,1} & a_{n,2} & \cdots & a_{n,n}

\end{vmatrix}.</math>

There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.

Leibniz formula

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3 × 3 matrices

The Leibniz formula for the determinant of a Template:Math matrix is the following:

<math>\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}

= aei + bfg + cdh - ceg - bdi - afh.\ </math>

In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example, bdi has b from the first row second column, d from the second row first column, and i from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and negative for an odd number. For the example of bdi, the single transposition of bd to db gives dbi, whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign.

File:Sarrus rule1.svg

Rule of Sarrus

The rule of Sarrus is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a Template:Math matrix does not carry over into higher dimensions.

n × n matrices

Generalizing the above to higher dimensions, the determinant of an <math>n \times n</math> matrix is an expression involving permutations and their signatures. A permutation of the set <math>\{1, 2, \dots, n \}</math> is a bijective function <math>\sigma</math> from this set to itself, with values <math>\sigma(1), \sigma(2),\ldots,\sigma(n)</math> exhausting the entire set. The set of all such permutations, called the symmetric group, is commonly denoted <math>S_n</math>. The signature <math>\sgn(\sigma)</math> of a permutation <math>\sigma</math> is <math>+1,</math> if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is <math>-1.</math>

Given a matrix

<math>A=\begin{bmatrix}

a_{1,1}\ldots a_{1,n}\\ \vdots\qquad\vdots\\ a_{n,1}\ldots a_{n,n} \end{bmatrix},</math> the Leibniz formula for its determinant is, using sigma notation for the sum,

<math>\det(A)=\begin{vmatrix}

a_{1,1}\ldots a_{1,n}\\ \vdots\qquad\vdots\\ a_{n,1}\ldots a_{n,n} \end{vmatrix} = \sum_{\sigma \in S_n}\sgn(\sigma)a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.</math>

Using pi notation for the product, this can be shortened into

<math>\det(A) = \sum_{\sigma \in S_n} \left( \sgn(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}\right)</math>.

The Levi-Civita symbol <math>\varepsilon_{i_1,\ldots,i_n}</math> is defined on the Template:Mvar-tuples of integers in <math>\{1,\ldots,n\}</math> as Template:Math if two of the integers are equal, and otherwise as the signature of the permutation defined by the n-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes

<math>\det(A) = \sum_{i_1,i_2,\ldots,i_n} \varepsilon_{i_1\cdots i_n} a_{1,i_1} \!\cdots a_{n,i_n},</math>

where the sum is taken over all Template:Mvar-tuples of integers in <math>\{1,\ldots,n\}.</math> <ref>Template:Cite book</ref><ref>Template:Harvnb</ref>

Properties

Characterization of the determinant

The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an <math>n \times n</math> matrix A as being composed of its <math>n</math> columns, so denoted as

<math>A = \big ( a_1, \dots, a_n \big ),</math>

where the column vector <math>a_i</math> (for each i) is composed of the entries of the matrix in the i-th column.

2. <math>\det\left(I\right) = 1</math>, where <math>I</math> is an identity matrix.
4. The determinant is multilinear: if the jth column of a matrix <math>A</math> is written as a linear combination <math>a_j = r \cdot v + w</math> of two column vectors v and w and a number r, then the determinant of A is expressible as a similar linear combination:

   <math>\begin{align}|A|

 &= \big | a_1, \dots, a_{j-1}, r \cdot v + w, a_{j+1}, \dots, a_n | \\
 &= r \cdot | a_1, \dots, v, \dots a_n | + | a_1, \dots, w, \dots, a_n |

\end{align}</math>

2. The determinant is alternating: whenever two columns of a matrix are identical, its determinant is 0:

   <math>| a_1, \dots, v, \dots, v, \dots, a_n| = 0.</math>

If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any <math>n \times n</math> matrix A a number that satisfies these three properties.<ref>Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191.</ref> This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.

To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.Template:Citation needed

Immediate consequences

These rules have several further consequences:

• The determinant is a homogeneous function, i.e., <math display="block">\det(cA) = c^n\det(A)</math> (for an <math>n \times n</math> matrix <math>A</math>).
• Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): <math display="block">|a_1, \dots, a_j, \dots a_i, \dots, a_n| = - |a_1, \dots, a_i, \dots, a_j, \dots, a_n|.</math> This formula can be applied iteratively when several columns are swapped. For example <math display="block">|a_3, a_1, a_2, a_4 \dots, a_n| = - |a_1, a_3, a_2, a_4, \dots, a_n| = |a_1, a_2, a_3, a_4, \dots, a_n|.</math> Yet more generally, any permutation of the columns multiplies the determinant by the sign of the permutation.
• If some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
• Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
• If <math>A</math> is a triangular matrix, i.e. <math>a_{ij}=0</math>, whenever <math>i>j</math> or, alternatively, whenever <math>i<j</math>, then its determinant equals the product of the diagonal entries: <math display="block">\det(A) = a_{11} a_{22} \cdots a_{nn} = \prod_{i=1}^n a_{ii}.</math> Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a diagonal matrix (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation <math>\sigma</math> which gives a non-zero contribution is the identity permutation.

Example

These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix <math>A</math> using that method:

<math>A = \begin{bmatrix}

-2 & -1 & 2 \\
 2 & 1 & 4 \\
-3 & 3 & -1

\end{bmatrix}. </math>

Computation of the determinant of matrix <math>A</math>

Matrix	<math>B = \begin{bmatrix} -3 & -1 & 2 \\ 3 & 1 & 4 \\ 0 & 3 & -1 \end{bmatrix} </math> || <math>C = \begin{bmatrix} -3 & 5 & 2 \\ 3 & 13 & 4 \\ 0 & 0 & -1 \end{bmatrix} </math>	<math>D = \begin{bmatrix} 5 & -3 & 2 \\ 13 & 3 & 4 \\ 0 & 0 & -1 \end{bmatrix} </math>	<math>E = \begin{bmatrix} 18 & -3 & 2 \\ 0 & 3 & 4 \\ 0 & 0 & -1 \end{bmatrix} </math>
Obtained by	add the second column to the first	add 3 times the third column to the second	swap the first two columns	add <math>-\frac{13} 3</math> times the second column to the first
Determinant	A| = |B|</math>	<math>|B| = |C|</math>	<math>|D| = -|C|</math>	<math>|E| = |D|</math>

Combining these equalities gives <math>|A| = -|E| = -(18 \cdot 3 \cdot (-1)) = 54.</math>

Transpose

The determinant of the transpose of <math>A</math> equals the determinant of A:

<math>\det\left(A^\textsf{T}\right) = \det(A)</math>.

This can be proven by inspecting the Leibniz formula.<ref>Template:Harvnb</ref> This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an Template:Math matrix as being composed of n rows, the determinant is an n-linear function.

Multiplicativity and matrix groups

The determinant is a multiplicative map, i.e., for square matrices <math>A</math> and <math>B</math> of equal size, the determinant of a matrix product equals the product of their determinants:

<math>\det(AB) = \det (A) \det (B)</math>

This key fact can be proven by observing that, for a fixed matrix <math>B</math>, both sides of the equation are alternating and multilinear as a function depending on the columns of <math>A</math>. Moreover, they both take the value <math>\det B</math> when <math>A</math> is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.<ref>Alternatively, Template:Harvnb proves this result using the functoriality of the exterior power.</ref>

A matrix <math>A</math> with entries in a field is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by

<math>\det\left(A^{-1}\right) = \frac{1}{\det(A)} = [\det(A)]^{-1}</math>.

In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size <math>n</math> over a field <math>K</math>) forms a group known as the general linear group <math>\operatorname{GL}_n(K)</math> (respectively, a subgroup called the special linear group <math>\operatorname{SL}_n(K) \subset \operatorname{GL}_n(K)</math>. More generally, the word "special" indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

Because the determinant respects multiplication and inverses, it is in fact a group homomorphism from <math>\operatorname{GL}_n(K)</math> into the multiplicative group <math>K^\times</math> of nonzero elements of <math>K</math>. This homomorphism is surjective and its kernel is <math>\operatorname{SL}_n(K)</math> (the matrices with determinant one). Hence, by the first isomorphism theorem, this shows that <math>\operatorname{SL}_n(K)</math> is a normal subgroup of <math>\operatorname{GL}_n(K)</math>, and that the quotient group <math>\operatorname{GL}_n(K)/\operatorname{SL}_n(K)</math> is isomorphic to <math>K^\times</math>.

The Cauchy–Binet formula is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

Laplace expansion

Laplace expansion expresses the determinant of a matrix <math>A</math> recursively in terms of determinants of smaller matrices, known as its minors. The minor <math>M_{i,j}</math> is defined to be the determinant of the <math>(n-1) \times (n-1)</math> matrix that results from <math>A</math> by removing the <math>i</math>-th row and the <math>j</math>-th column. The expression <math>(-1)^{i+j}M_{i,j}</math> is known as a cofactor. For every <math>i</math>, one has the equality

<math>\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{i,j} M_{i,j},</math>

which is called the Laplace expansion along the Template:Mvarth row. For example, the Laplace expansion along the first row (<math>i=1</math>) gives the following formula:

<math>

 \begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix} =
 a\begin{vmatrix}e&f\\ h&i\end{vmatrix} - b\begin{vmatrix}d&f\\ g&i\end{vmatrix} + c\begin{vmatrix}d&e\\ g&h\end{vmatrix}

</math> Unwinding the determinants of these <math>2 \times 2</math>-matrices gives back the Leibniz formula mentioned above. Similarly, the Laplace expansion along the <math>j</math>-th column is the equality

<math>\det(A)= \sum_{i=1}^n (-1)^{i+j} a_{i,j} M_{i,j}.</math>

Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the Vandermonde matrix <math display="block">\begin{vmatrix}

           1 &         1 &         1 & \cdots &         1 \\
         x_1 &       x_2 &       x_3 & \cdots &       x_n \\
       x_1^2 &     x_2^2 &     x_3^2 & \cdots &     x_n^2 \\
      \vdots &    \vdots &    \vdots & \ddots &    \vdots \\
   x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \cdots & x_n^{n-1}
 \end{vmatrix} =
 \prod_{1 \leq i < j \leq n} \left(x_j - x_i\right).

</math>The n-term Laplace expansion along a row or column can be generalized to write an n x n determinant as a sum of <math>\tbinom nk</math> terms, each the product of the determinant of a k x k submatrix and the determinant of the complementary (n−k) x (n−k) submatrix.

Adjugate matrix

The adjugate matrix <math>\operatorname{adj}(A)</math> is the transpose of the matrix of the cofactors, that is,

<math>(\operatorname{adj}(A))_{i,j} = (-1)^{i+j} M_{ji}.</math>

For every matrix, one has<ref>Template:Harvnb.</ref>

<math>(\det A) I = A\operatorname{adj}A = (\operatorname{adj}A)\,A. </math>

Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix:

<math>A^{-1} = \frac 1{\det A}\operatorname{adj}A. </math>

Block matrices

The formula for the determinant of a <math>2 \times 2</math> matrix above continues to hold, under appropriate further assumptions, for a block matrix, i.e., a matrix composed of four submatrices <math>A, B, C, D</math> of dimension <math>m \times m</math>, <math>m \times n</math>, <math>n \times m</math> and <math>n \times n</math>, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is

<math>\det\begin{pmatrix}A& 0\\ C& D\end{pmatrix} = \det(A) \det(D) = \det\begin{pmatrix}A& B\\ 0& D\end{pmatrix}.</math>

If <math>A</math> is invertible, then it follows with results from the section on multiplicativity that

<math>\begin{align}

\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} & = \det(A)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} \underbrace{\det\begin{pmatrix}A^{-1}& -A^{-1} B\\ 0& I_n\end{pmatrix}}_{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\ & = \det(A) \det\begin{pmatrix}I_m& 0\\ C A^{-1}& D-C A^{-1} B\end{pmatrix}\\ & = \det(A) \det(D - C A^{-1} B), \end{align}</math> which simplifies to <math>\det (A) (D - C A^{-1} B)</math> when <math>D</math> is a <math>1 \times 1</math> matrix.

A similar result holds when <math>D</math> is invertible, namely

<math>\begin{align}

\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} & = \det(D)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} \underbrace{\det\begin{pmatrix}I_m& 0\\ -D^{-1} C& D^{-1}\end{pmatrix}}_{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\ & = \det(D) \det\begin{pmatrix}A - B D^{-1} C& B D^{-1}\\ 0& I_n\end{pmatrix}\\ & = \det(D) \det(A - B D^{-1} C). \end{align}</math> Both results can be combined to derive Sylvester's determinant theorem, which is also stated below.

If the blocks are square matrices of the same size further formulas hold. For example, if <math>C</math> and <math>D</math> commute (i.e., <math>CD=DC</math>), then<ref>Template:Cite journal</ref>

<math>\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(AD - BC).</math>

This formula has been generalized to matrices composed of more than <math>2 \times 2</math> blocks, again under appropriate commutativity conditions among the individual blocks.<ref>Template:Cite journal</ref>

For <math>A = D</math> and <math>B = C</math>, the following formula holds (even if <math>A</math> and <math>B</math> do not commute).

<math>\det\begin{pmatrix}A & B\\ B & A\end{pmatrix} = \det\begin{pmatrix}A+B & B\\ B+A & A\end{pmatrix} = \det\begin{pmatrix}A+B & B\\ 0 & A-B\end{pmatrix} = \det(A+B) \det(A-B).</math>

Sylvester's determinant theorem

Sylvester's determinant theorem states that for A, an Template:Math matrix, and B, an Template:Math matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix):

<math>\det\left(I_\mathit{m} + AB\right) = \det\left(I_\mathit{n} + BA\right),</math>

where I_m and I_n are the Template:Math and Template:Math identity matrices, respectively.

From this general result several consequences follow. Template:Ordered list

A generalization is <math>\det\left(Z + AWB\right) = \det\left( Z\right) \det\left(W \right) \det\left(W^{-1} + B Z^{-1} A\right)</math>(see Matrix determinant lemma), where Z is an Template:Math invertible matrix and W is an Template:Math invertible matrix.

Sum

The determinant of the sum <math>A+B</math> of two square matrices of the same size is not in general expressible in terms of the determinants of A and of B.

However, for positive semidefinite matrices <math>A</math>, <math>B</math> and <math>C</math> of equal size, <math display=block>\det(A + B + C) + \det(C) \geq \det(A + C) + \det(B + C)\text{,}</math> with the corollary<ref>Template:Cite arXiv</ref><ref>Template:Cite journal</ref> <math display=block>\det(A + B) \geq \det(A) + \det(B)\text{.}</math>

Brunn–Minkowski theorem implies that the Template:Mvarth root of determinant is a concave function, when restricted to Hermitian positive-definite <math>n\times n</math> matrices.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Therefore, if Template:Mvar and Template:Mvar are Hermitian positive-definite <math>n\times n</math> matrices, one has <math display=block>\sqrt[n]{\det(A+B)}\geq\sqrt[n]{\det(A)}+\sqrt[n]{\det(B)},</math> since the Template:Mvarth root of the determinant is a homogeneous function.

Sum identity for 2×2 matrices

For the special case of <math>2\times 2</math> matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:

<math>\det(A+B) = \det(A) + \det(B) + \text{tr}(A)\text{tr}(B) - \text{tr}(AB).</math>

Properties of the determinant in relation to other notions

Eigenvalues and characteristic polynomial

The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial of a matrix. Let <math>A</math> be an <math>n \times n</math> matrix with complex entries. Then, by the Fundamental Theorem of Algebra, <math>A</math> must have exactly n eigenvalues <math>\lambda_1, \lambda_2, \ldots, \lambda_n</math>. (Here it is understood that an eigenvalue with algebraic multiplicity Template:Mvar occurs Template:Mvar times in this list.) Then, it turns out the determinant of Template:Mvar is equal to the product of these eigenvalues,

<math>\det(A) = \prod_{i=1}^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n.</math>

The product of all non-zero eigenvalues is referred to as pseudo-determinant.

From this, one immediately sees that the determinant of a matrix <math>A</math> is zero if and only if <math>0</math> is an eigenvalue of <math>A</math>. In other words, <math>A</math> is invertible if and only if <math>0</math> is not an eigenvalue of <math>A</math>.

The characteristic polynomial is defined as<ref>Template:Harvnb, Template:Harvnb</ref>

<math>\chi_A(t) = \det(t \cdot I - A).</math>

Here, <math>t</math> is the indeterminate of the polynomial and <math>I</math> is the identity matrix of the same size as <math>A</math>. By means of this polynomial, determinants can be used to find the eigenvalues of the matrix <math>A</math>: they are precisely the roots of this polynomial, i.e., those complex numbers <math>\lambda</math> such that

<math>\chi_A(\lambda) = 0.</math>

A Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices

<math>A_k := \begin{bmatrix}

 a_{1,1} & a_{1,2} & \cdots & a_{1,k} \\
 a_{2,1} & a_{2,2} & \cdots & a_{2,k} \\
  \vdots &  \vdots & \ddots &  \vdots \\
 a_{k,1} & a_{k,2} & \cdots & a_{k,k}

\end{bmatrix}</math>

being positive, for all <math>k</math> between <math>1</math> and <math>n</math>.<ref>Template:Harvnb</ref>

Trace

The trace tr(A) is by definition the sum of the diagonal entries of Template:Mvar and also equals the sum of the eigenvalues. Thus, for complex matrices Template:Mvar,

<math>\det(\exp(A)) = \exp(\operatorname{tr}(A))</math>

or, for real matrices Template:Mvar,

<math>\operatorname{tr}(A) = \log(\det(\exp(A))).</math>

Here exp(Template:Mvar) denotes the matrix exponential of Template:Mvar, because every eigenvalue Template:Mvar of Template:Mvar corresponds to the eigenvalue exp(Template:Mvar) of exp(Template:Mvar). In particular, given any logarithm of Template:Mvar, that is, any matrix Template:Mvar satisfying

<math>\exp(L) = A</math>

the determinant of Template:Mvar is given by

<math>\det(A) = \exp(\operatorname{tr}(L)).</math>

For example, for Template:Math, Template:Math, and Template:Math, respectively,

<math>\begin{align}

 \det(A) &= \frac{1}{2}\left(\left(\operatorname{tr}(A)\right)^2 -  \operatorname{tr}\left(A^2\right)\right), \\
 \det(A) &= \frac{1}{6}\left(\left(\operatorname{tr}(A)\right)^3 - 3\operatorname{tr}(A) ~ \operatorname{tr}\left(A^2\right) + 2 \operatorname{tr}\left(A^3\right)\right), \\
 \det(A) &= \frac{1}{24}\left(\left(\operatorname{tr}(A)\right)^4 - 6\operatorname{tr}\left(A^2\right)\left(\operatorname{tr}(A)\right)^2 + 3\left(\operatorname{tr}\left(A^2\right)\right)^2 + 8\operatorname{tr}\left(A^3\right)~\operatorname{tr}(A) - 6\operatorname{tr}\left(A^4\right)\right).

\end{align}</math>

cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic Template:Mvar, Template:Math the signed constant term of the characteristic polynomial, determined recursively from

<math>c_n = 1; ~~~c_{n-m} = -\frac{1}{m}\sum_{k=1}^m c_{n-m+k} \operatorname{tr}\left(A^k\right) ~~(1 \le m \le n)~.</math>

In the general case, this may also be obtained from<ref>A proof can be found in the Appendix B of Template:Cite journal</ref>

<math>\det(A) = \sum_{\begin{array}{c}k_1,k_2,\ldots,k_n \geq 0\\k_1+2k_2+\cdots+nk_n=n\end{array}}\prod_{l=1}^n \frac{(-1)^{k_l+1}}{l^{k_l}k_l!} \operatorname{tr}\left(A^l\right)^{k_l},</math>

where the sum is taken over the set of all integers Template:Math satisfying the equation

<math>\sum_{l=1}^n lk_l = n.</math>

The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments s_l = −(l – 1)! tr(A^l) as

<math>\det(A) = \frac{(-1)^n}{n!} B_n(s_1, s_2, \ldots, s_n).</math>

This formula can also be used to find the determinant of a matrix Template:Math with multidimensional indices Template:Math and Template:Math. The product and trace of such matrices are defined in a natural way as

<math>(AB)^I_J = \sum_K A^I_K B^K_J, \operatorname{tr}(A) = \sum_I A^I_I.</math>

An important arbitrary dimension Template:Mvar identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value,

<math>\det(I + A) = \sum_{k=0}^\infty \frac{1}{k!} \left(-\sum_{j=1}^\infty \frac{(-1)^j}{j} \operatorname{tr}\left(A^j\right)\right)^k\,,</math>

where Template:Math is the identity matrix. More generally, if

<math>\sum_{k=0}^\infty \frac{1}{k!} \left(-\sum_{j=1}^\infty \frac{(-1)^j s^j}{j}\operatorname{tr}\left(A^j\right)\right)^k\,,</math>

is expanded as a formal power series in Template:Mvar then all coefficients of Template:Mvar^{Template:Mvar} for Template:Math are zero and the remaining polynomial is Template:Math.

Upper and lower bounds

For a positive definite matrix Template:Math, the trace operator gives the following tight lower and upper bounds on the log determinant

<math>\operatorname{tr}\left(I - A^{-1}\right) \le \log\det(A) \le \operatorname{tr}(A - I)</math>

with equality if and only if Template:Math. This relationship can be derived via the formula for the Kullback-Leibler divergence between two multivariate normal distributions.

Also,

<math>\frac{n}{\operatorname{tr}\left(A^{-1}\right)} \leq \det(A)^\frac{1}{n} \leq \frac{1}{n}\operatorname{tr}(A) \leq \sqrt{\frac{1}{n}\operatorname{tr}\left(A^2\right)}.</math>

These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.

Derivative

The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from <math>\mathbf R^{n \times n}</math> to <math>\mathbf R</math>. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:<ref>Template:Harvnb</ref>

<math>\frac{d \det(A)}{d \alpha} = \operatorname{tr}\left(\operatorname{adj}(A) \frac{d A}{d \alpha}\right).</math>

where <math>\operatorname{adj}(A)</math> denotes the adjugate of <math>A</math>. In particular, if <math>A</math> is invertible, we have

<math>\frac{d \det(A)}{d \alpha} = \det(A) \operatorname{tr}\left(A^{-1} \frac{d A}{d \alpha}\right).</math>

Expressed in terms of the entries of <math>A</math>, these are

<math> \frac{\partial \det(A)}{\partial A_{ij}}= \operatorname{adj}(A)_{ji} = \det(A)\left(A^{-1}\right)_{ji}.</math>

Yet another equivalent formulation is

<math>\det(A + \epsilon X) - \det(A) = \operatorname{tr}(\operatorname{adj}(A) X) \epsilon + O\left(\epsilon^2\right) = \det(A) \operatorname{tr}\left(A^{-1} X\right) \epsilon + O\left(\epsilon^2\right)</math>,

using big O notation. The special case where <math>A = I</math>, the identity matrix, yields

<math>\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O\left(\epsilon^2\right).</math>

This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group <math>\operatorname{SL}_n</math> is defined by the equation <math>\det A = 1</math>. The above formula shows that its Lie algebra is the special linear Lie algebra <math>\mathfrak{sl}_n</math> consisting of those matrices having trace zero.

Writing a <math>3 \times 3</math> matrix as <math>A = \begin{bmatrix}a & b & c\end{bmatrix}</math> where <math>a, b,c</math> are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two:

<math>\begin{align}

 \nabla_\mathbf{a}\det(A) &= \mathbf{b} \times \mathbf{c} \\
 \nabla_\mathbf{b}\det(A) &= \mathbf{c} \times \mathbf{a} \\
 \nabla_\mathbf{c}\det(A) &= \mathbf{a} \times \mathbf{b}.

\end{align}</math>

History

Historically, determinants were used long before matrices: A determinant was originally defined as a property of a system of linear equations. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.<ref>Template:Harvnb</ref>

Determinants proper originated separately from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693.<ref>Cajori, F. A History of Mathematics p. 80</ref><ref name="Campbell" /><ref>Template:Harvnb</ref><ref>A Brief History of Linear Algebra and Matrix Theory at: {{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:Harvtxt stated, without proof, Cramer's rule.<ref>Template:Harvnb</ref> Both Cramer and also Template:Harvtxt were led to determinants by the question of plane curves passing through a given set of points.<ref>Template:Harvtxt</ref>

Vandermonde (1771) first recognized determinants as independent functions.<ref name="Campbell">Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971</ref> Template:Harvtxt gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case.<ref>Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development [London, England: Macmillan and Co., Ltd., 1906]. Template:JFM</ref> Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.

Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a quadratic form.<ref>Template:Harvnb</ref> Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.Template:Clarify

The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of Template:Math reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word "determinant" in its present sense,<ref>The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the Journal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29–112 (1815).</ref><ref>Origins of mathematical terms: http://jeff560.tripod.com/d.html</ref> summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's.<ref name="Campbell" /><ref>History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html</ref> With him begins the theory in its generality.

Template:Harvtxt used the functional determinant which Sylvester later called the Jacobian.<ref>Template:Harvnb</ref> In his memoirs in Crelle's Journal for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi's last memoirs, Sylvester (1839) and Cayley began their work. Template:Harvnb introduced the modern notation for the determinant using vertical bars.<ref>Template:Harvnb</ref><ref>History of matrix notation: http://jeff560.tripod.com/matrices.html</ref>

The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

Applications

Cramer's rule

Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as <math>Ax = b</math>. This equation has a unique solution <math>x</math> if and only if <math>\det (A)</math> is nonzero. In this case, the solution is given by Cramer's rule:

<math>x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, 2, 3, \ldots, n</math>

where <math>A_i</math> is the matrix formed by replacing the <math>i</math>-th column of <math>A</math> by the column vector <math>b</math>. This follows immediately by column expansion of the determinant, i.e.

<math>\det(A_i) =

 \det\begin{bmatrix}a_1 & \ldots & b & \ldots & a_n\end{bmatrix} 

</math> <math>

 =\sum_{j=1}^n x_j\det\begin{bmatrix}a_1 & \ldots & a_{i-1} & a_j & a_{i+1} & \ldots & a_n\end{bmatrix} =
 x_i\det(A)

</math>

where the vectors <math>a_j</math> are the columns of A. The rule is also implied by the identity

<math>A\, \operatorname{adj}(A) = \operatorname{adj}(A)\, A = \det(A)\, I_n.</math>

Cramer's rule can be implemented in <math>\operatorname O(n^3)</math> time, which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.<ref>Template:Harvnb</ref>

Linear independence

Determinants can be used to characterize linearly dependent vectors: <math>\det A</math> is zero if and only if the column vectors of the matrix <math>A</math> are linearly dependent.<ref>Template:Harvnb</ref> For example, given two linearly independent vectors <math>v_1, v_2 \in \mathbf R^3</math>, a third vector <math>v_3</math> lies in the plane spanned by the former two vectors exactly if the determinant of the <math>3 \times 3</math> matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given functions <math>f_1(x), \dots, f_n(x)</math> (supposed to be <math>n-1</math> times differentiable), the Wronskian is defined to be

<math>W(f_1, \ldots, f_n)(x) =

 \begin{vmatrix}
           f_1(x) &         f_2(x) & \cdots &         f_n(x) \\
          f_1'(x) &        f_2'(x) & \cdots &        f_n'(x) \\
           \vdots &         \vdots & \ddots &         \vdots \\
   f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)

\end{vmatrix}.</math>

It is non-zero (for some <math>x</math>) in a specified interval if and only if the given functions and all their derivatives up to order <math>n-1</math> are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root.<ref>Template:Harvnb</ref>

Orientation of a basis

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The determinant can be thought of as assigning a number to every sequence of n vectors in Rⁿ, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a basis for Rⁿ. In that case, the sign of the determinant determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. In the case of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an orthogonal matrix with entries in Rⁿ represents an orthonormal basis in Euclidean space, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.

More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal Template:Math or Template:Math matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if <math>f : \mathbf R^n \to \mathbf R^n</math> is the linear map given by multiplication with a matrix <math>A</math>, and <math>S \subset \mathbf R^n</math> is any measurable subset, then the volume of <math>f(S)</math> is given by <math>|\det(A)|</math> times the volume of <math>S</math>.<ref>Template:Harvnb</ref> More generally, if the linear map <math>f : \mathbf R^n \to \mathbf R^m</math> is represented by the <math>m \times n</math> matrix <math>A</math>, then the <math>n</math>-dimensional volume of <math>f(S)</math> is given by:

<math>\operatorname{volume}(f(S)) = \sqrt{\det\left(A^\textsf{T} A\right)} \operatorname{volume}(S).</math>

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices <math>a, b, c, d</math>, <math>\frac 1 6 \cdot |\det(a-b,b-c,c-d)|</math>, or any other combination of pairs of vertices that form a spanning tree over the vertices.

File:Jacobian determinant and distortion.svg

A nonlinear map <math>f \colon \mathbf{R}^{2} \to \mathbf{R}^{2}</math> sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.

For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For

<math>f: \mathbf R^n \rightarrow \mathbf R^n,</math>

the Jacobian matrix is the Template:Math matrix whose entries are given by the partial derivatives

<math>D(f) = \left(\frac {\partial f_i}{\partial x_j}\right)_{1 \leq i, j \leq n}.</math>

Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of Rⁿ (the domain of f), the integral over f(U) of some other function Template:Math is given by

<math>\int_{f(U)} \phi(\mathbf{v})\, d\mathbf{v} = \int_U \phi(f(\mathbf{u})) \left|\det(\operatorname{D}f)(\mathbf{u})\right| \,d\mathbf{u}.</math>

The Jacobian also occurs in the inverse function theorem.

When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.<ref>Template:Cite book</ref>

Abstract algebraic aspects Template:Anchor

Determinant of an endomorphism

The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that Template:Math. Indeed, repeatedly applying the above identities yields

<math>\det(A) = \det(X)^{-1} \det(B)\det(X) = \det(B) \det(X)^{-1} \det(X) = \det(B).</math>

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation

<math>T : V \to V</math>

for some finite-dimensional vector space V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

Square matrices over commutative rings

The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a commutative ring <math>R</math>, such as the integers <math>\mathbf Z</math>, as opposed to the field of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies <math>\det(I) = 1</math> still holds, as do all the properties that result from that characterization.<ref>Template:Harvnb</ref>

A matrix <math>A \in \operatorname{Mat}_{n \times n}(R)</math> is invertible (in the sense that there is an inverse matrix whose entries are in <math>R</math>) if and only if its determinant is an invertible element in <math>R</math>.<ref>Template:Harvnb</ref> For <math>R = \mathbf Z</math>, this means that the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant being multiplicative, it defines a group homomorphism

<math>\operatorname{GL}_n(R) \rightarrow R^\times, </math>

between the general linear group (the group of invertible <math>n \times n</math>-matrices with entries in <math>R</math>) and the multiplicative group of units in <math>R</math>. Since it respects the multiplication in both groups, this map is a group homomorphism.

File:Determinant as a natural transformation.svg

The determinant is a natural transformation.

Given a ring homomorphism <math>f : R \to S</math>, there is a map <math>\operatorname{GL}_n(f) : \operatorname{GL}_n(R) \to \operatorname{GL}_n(S)</math> given by replacing all entries in <math>R</math> by their images under <math>f</math>. The determinant respects these maps, i.e., the identity

<math>f(\det((a_{i,j}))) = \det ((f(a_{i,j})))</math>

holds. In other words, the displayed commutative diagram commutes.

For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo <math>m</math> of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo <math>m</math> (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation between the two functors <math>\operatorname{GL}_n</math> and <math>(-)^\times</math>.<ref>Template:Harvnb. See also Template:Section link.</ref> Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,

<math>\det: \operatorname{GL}_n \to \mathbb G_m.</math>

Exterior algebra

The determinant of a linear transformation <math>T : V \to V</math> of an <math>n</math>-dimensional vector space <math>V</math> or, more generally a free module of (finite) rank <math>n</math> over a commutative ring <math>R</math> can be formulated in a coordinate-free manner by considering the <math>n</math>-th exterior power <math>\bigwedge^n V</math> of <math>V</math>.<ref>Template:Harvnb</ref> The map <math>T</math> induces a linear map

<math>\begin{align}

           \bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\
 v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n.

\end{align}</math>

As <math>\bigwedge^n V</math> is one-dimensional, the map <math>\bigwedge^n T</math> is given by multiplying with some scalar, i.e., an element in <math>R</math>. Some authors such as Template:Harv use this fact to define the determinant to be the element in <math>R</math> satisfying the following identity (for all <math>v_i \in V</math>):

<math>\left(\bigwedge^n T\right)\left(v_1 \wedge \dots \wedge v_n\right) = \det(T) \cdot v_1 \wedge \dots \wedge v_n.</math>

This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on <math>n</math>-tuples of vectors in <math>R^n</math>. For this reason, the highest non-zero exterior power <math>\bigwedge^n V</math> (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of <math>V</math> and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms <math>\bigwedge^k V</math> with <math>k < n</math>.<ref>Template:Harvnb, Template:Harvnb</ref>

Berezin integral

The conventional definition of the determinant, as a sum over permutations over a product of matrix elements, can be written using the somewhat surprising notation of the Berezin integral. In this notation, the determinant can be written as

<math>\int \exp\left[-\theta^TA\eta\right] \,d\theta\,d\eta = \det A </math>

This holds for any <math>n\times n</math>-dimensional matrix <math>A.</math> The symbols <math>\theta,\eta</math> are two <math>n</math>-dimensional vectors of anti-commuting Grassmann numbers (aka "supernumbers"), taken from the Grassmann algebra. The <math>\exp</math> here is the exponential function. The integral sign is meant to be understood as the Berezin integral. Despite the use of the integral symbol, this expression is in fact an entirely finite sum.

This unusual-looking expression can be understood as a notational trick that rewrites the conventional expression for the determinant

<math>\det A = \sum_{\sigma \in S_n}\sgn(\sigma)a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.</math>

by using some novel notation. The anti-commuting property of the Grassmann numbers captures the sign (signature) of the permutation, while the integral combined with the <math>\exp</math> ensures that all permutations are explored. That is, the Taylor's series for <math>\exp</math> terminates after exactly <math>n</math> terms, because the square of a Grassmann number is zero, and there are exactly <math>n</math> distinct Grassmann variables. Meanwhile, the integral is defined to vanish, if the corresponding Grassmann number does not appear in the integrand. Thus, the integral selects out only those terms in the <math>\exp</math> series that have exactly <math>n</math> distinct variables; all lower-order terms vanish. Thus, the somewhat magical combination of the integral sign, the use of anti-commuting variables, and the Taylor's series for <math>\exp</math> just encodes a finite sum, identical to the conventional summation.

This form is popular in physics, where it is often used as a stand-in for the Jacobian determinant. The appeal is that, notationally, the integral takes the form of a path integral, such as in the path integral formulation for quantized Hamiltonian mechanics. An example can be found in the theory of Fadeev–Popov ghosts; although this theory may seem rather abstruse, it's best to keep in mind that the use of the ghost fields is little more than a notational trick to express a Jacobian determinant.

The Pfaffian <math>\mathrm{Pf}\,A</math> of a skew-symmetric matrix <math>A</math> is the square-root of the determinant: that is, <math>\left(\mathrm{Pf}\,A\right)^2=\det A.</math> The Berezin integral form for the Pfaffian is even more suggestive; it is

<math>\int \exp\left[- \tfrac{1}{2} \theta^T A \theta\right] \,d\theta = \mathrm{Pf}\, A </math>

The integrand has exactly the same formal structure as a normal Gaussian distribution, albeit with Grassman numbers, instead of real numbers. This formal resemblance accounts for the occasional appearance of supernumbers in the theory of stochastic dynamics and stochastic differential equations.

Generalizations and related notions

Determinants as treated above admit several variants: the permanent of a matrix is defined as the determinant, except that the factors <math>\sgn(\sigma)</math> occurring in Leibniz's rule are omitted. The immanant generalizes both by introducing a character of the symmetric group <math>S_n</math> in Leibniz's rule.

Determinants for finite-dimensional algebras

For any associative algebra <math>A</math> that is finite-dimensional as a vector space over a field <math>F</math>, there is a determinant map <ref>Template:Harvnb</ref>

<math>\det : A \to F.</math>

This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the matrix algebra <math>A = \operatorname{Mat}_{n \times n}(F)</math>, but also includes several further cases including the determinant of a quaternion,

<math>\det (a + ib+jc+kd) = a^2 + b^2 + c^2 + d^2</math>,

the norm <math>N_{L/F} : L \to F</math> of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of a central simple algebra, also arise as special cases of this construction.

Infinite matrices

For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.

The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula

<math>\det(I+A) = \exp(\operatorname{tr}(\log(I+A))). </math>

Another infinite-dimensional notion of determinant is the functional determinant.

Operators in von Neumann algebras

For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.

Related notions for non-commutative rings

For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for Template:Math,<ref>In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars a, b: <math display=block>\begin{align} ab

 &= ab \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}
  =  a \begin{vmatrix}1&0 \\ 0&b\end{vmatrix} \\[5mu]
 &=    \begin{vmatrix}a&0 \\ 0&b\end{vmatrix} 
  =  b \begin{vmatrix}a&0 \\ 0&1\end{vmatrix}
  = ba \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}
  = ba,

\end{align}</math>

a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.</ref> so there is no good definition of the determinant in this setting.

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero Template:Clarify span with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of <math>\mathbb Z_2</math>-graded rings).<ref>Template:Citation</ref> Manin matrices form the class closest to matrices with commutative elements.

Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.<ref>"... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms.", see Template:Harvnb.</ref> Computational geometry, however, does frequently use calculations related to determinants.<ref>Template:Harvnb</ref>

While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating <math>n!</math> (<math>n</math> factorial) products for an <math>n \times n</math> matrix. Thus, the number of required operations grows very quickly: it is of order <math>n!</math>. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.

Gaussian elimination

Gaussian elimination consists of left multiplying a matrix by elementary matrices for getting a matrix in a row echelon form. One can restrict the computation to elementary matrices of determinant Template:Math. In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a triangular matrix, its determinant is the product of the entries of its diagonal.

So, the determinant can be computed for almost free from the result of a Gaussian elimination.

Decomposition methods

Some methods compute <math>\det(A)</math> by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite matrices). These methods are of order <math>\operatorname O(n^3)</math>, which is a significant improvement over <math>\operatorname O (n!)</math>.<ref>Template:Cite arXiv</ref>

For example, LU decomposition expresses <math>A</math> as a product

<math> A = PLU. </math>

of a permutation matrix <math>P</math> (which has exactly a single <math>1</math> in each column, and otherwise zeros), a lower triangular matrix <math>L</math> and an upper triangular matrix <math>U</math>. The determinants of the two triangular matrices <math>L</math> and <math>U</math> can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of <math>P</math> is just the sign <math>\varepsilon</math> of the corresponding permutation (which is <math>+1</math> for an even number of permutations and is <math> -1 </math> for an odd number of permutations). Once such a LU decomposition is known for <math>A</math>, its determinant is readily computed as

<math> \det(A) = \varepsilon \det(L)\cdot\det(U). </math>

Further methods

The order <math>\operatorname O(n^3)</math> reached by decomposition methods has been improved by different methods. If two matrices of order <math>n</math> can be multiplied in time <math>M(n)</math>, where <math>M(n) \ge n^a</math> for some <math>a>2</math>, then there is an algorithm computing the determinant in time <math>O(M(n))</math>.<ref>Template:Harvnb</ref> This means, for example, that an <math>\operatorname O(n^{2.376})</math> algorithm for computing the determinant exists based on the Coppersmith–Winograd algorithm. This exponent has been further lowered, as of 2016, to 2.373.<ref>Template:Harvnb</ref>

In addition to the complexity of the algorithm, further criteria can be used to compare algorithms. Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity <math>\operatorname O(n^4)</math> is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called closed ordered walks, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule.<ref>Template:Harvnb</ref> Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) method is of order <math>\operatorname O(n^3)</math>, but the bit length of intermediate values can become exponentially long.<ref>Template:Cite conference</ref> By comparison, the Bareiss Algorithm, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times <math>n</math>.<ref>Template:Harvnb, Template:Harvnb</ref>

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of Template:Math, where u and v are column vectors.

Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.<ref>Template:Cite journal</ref>

See also

• Cauchy determinant
• Cayley–Menger determinant
• Dieudonné determinant
• Slater determinant
• Determinantal conjecture

References

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• G. Baley Price (1947) "Some identities in the theory of determinants", American Mathematical Monthly 54:75–90 Template:Mr
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Historical references

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• Robert Forsyth Scott (1880): A Treatise on the Theory of Determinants and Their Applications in Analysis and Geometry, Cambridge University Press
• E. R. Hedrick: On Three Dimensional Determinants, Annals of Mathematics, Vol.1, No.1/4 (1899-1900), pp.49-67 (19pages). https://doi.org/10.2307/1967268 # Note: This is not the ordinal determinant.

External links

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• Template:MacTutor
• Determinant Interactive Program and Tutorial
• Linear algebra: determinants. Template:Webarchive Compute determinants of matrices up to order 6 using Laplace expansion you choose.
• Determinant Calculator Calculator for matrix determinants, up to the 8th order.
• Matrices and Linear Algebra on the Earliest Uses Pages
• Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.

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